Rajasthan Technical University 2011-2nd Sem B.Tech Information Technology I (Main & Back),- , Mathematics - III (3IT1) - Question Paper
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Roll No. {Total No. of Pages :| 4) | ||
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B.Tech. Illrd Semester ( Computer Engineei 3IT1 & 3C |
3E2071 kMain/Bac] ing & Info :S1 Mathei |
k) Examination, Feb. - 2011 rmation Technology natics - III | |
Time : 3 Hours Maximum Marks : 80 Min. Passing Marks : 24
Instructions to Candidates:
Attempt any five questions, selecting one question from each unit. All questions carry equal marks. (Schematic diagrams must be shown wherever necessary. Any data you feel missing may suitably be assumed and stated clearly. Units of quantities used/calculated must be stated clearly.)
Use of following supporting material is permitted during examination (Mentioned in form No. 205).
1. Graph Paper 2. Normal distribution Table.
Unit-I
1. a) Assuming that the petrol burnt (per hour) in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against
C km/hr.
a current of C km/hr in
b) State Kuhn - Tukkar condition and use K.T. conditions to Min. Z=f(x,yiz) = x2+y1+zz + 2Qx+10 y
S.t. * >40 '
x+y> 80 x+y+z> 120
a) Find the optimum solution of the following constraint multivariable problem. Min z = xf + (x2 +1)2 + (x3 -1)2
S.t. x,+5x2-3x3=6
b) Minimize /(*) = - (*,2 + )
S-t g,(x) = x,-x2 =0
and g2(x) = x, + x2 + x3 -1 = 0
by Lagrange multiplier method.
2. a) A firm manufactures headache pills in two sizes A and B. Size A contains 2
grains of aspirin, 5 grains of bicarbonate and 1 grain of codeine ; size B contains 1 grain of aspirin, 8 grains bicarbonate and 6 grains of codeine. It has been found by users that it require at least 12 grains of aspirin, 74 grains of bicarbonate and 12 grains of codeine for providing immediate effects. Determine graphically the least number of pills a patient should have to get immediate relief,
b) Solve the following L.P.P.
Min. z = x, - 3x2 + 2x3
S.t 3jc, - x, + 3x3 < 7 2x} + 4x2 < 12 -4xt + 3x2 + 8x3 < 10 and xp x2, x3 > 0
a) Use duality to solve the following LPP :
Min Z = 3x, + x2
S.t. Xj + x, > 1 2xj + 3x2 > 2 and x,, x2 > 0
b) A company is spending Rs. 1000 on transportation of its units to four warehouses from three factories. What can be the maximum saving by optimal scheduling. Solve the following transportation problem.
Factory <-Warehouses -> Factory
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W1 |
w2 |
W, |
w4 |
Capacity |
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19 |
30 |
50 |
10 |
7 |
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70 |
30 |
40 |
60 |
9 |
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40 |
8 |
70 |
20 |
18 |
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5 |
8 |
7 |
14 |
34 |
1
Warehouse Requirement
3. a) Find the sequence that minimize the total elapsed time required to complete
the following jobs on two machines Mj and Mr Jobs tABCDEF GH1 M,:254968754 M2 : 6 8 7 4 3 9 3 8 11
b) Use graphical method to minimize the time needed to process the following jobs on the machines shown below. For each machine find the job which should be done first. Also find the total elapsed time to complete both the jobs. *
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Job 1 : Sequence of machines |
A |
B |
C |
D |
E |
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Time (hrs) |
3 |
4 |
2 |
6 |
2 |
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Job 2 : Sequence of machines |
B |
C |
A |
D |
E |
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Time (hrs) |
5 OR |
4 |
3 |
2 |
6 |
a) A project schedule has the following characteristic :
Activity: 1-2 1-3 24 34 3-5 4-9 56 5-7 6-8 7-8 8-10 9-10
Time(days): 411165 48 1257
i) Construct a network diagram.
ii) Compute the earliest event time and Latest event time
iii) Determine the critical path and total project duration
iv) Compute total float and free float for each activity.
b) A project has a following time estimates :
Activity Estimated durations (days)
(i,j) Optimistic (to) Most likely (tm) Pessimistic (tp)
(1.2) 1 ' 1 7
(1.3) 1 4 7
(1.4) 2 2 8
(2.5) 1 1 1
(3.5) 2 5 14
(4.6) 2 5 8
(5.6) 3 6 15
i) Draw the project network.
ii) Find the expected duration and variance of each activity.
iii) Find the early and late occurrence time for each event and the expected project length.
iv) Calculate the variance and standard deviations of project length.
v) What is the probability that the project will be completed 4 days earlier than expected?
4. a) Find the Laplace transform of and find the value of f dt .
Jo t2
d2y - dy
o 1 -h 2--Yy t
b) Solve dt~ dt
given thatj>(0) =-3, ><1) = 1
$
a) Use convolution theorem to find the inverse Laplace transform of (j2 + )2
b) Solve = 24 where u = u (x,t)
, dt ox
B.C : u (0, t) = 0 = u (5, t) and u (1,0) = 10 Sin4vx
Unit-V
5. a) Given:
e : 0 5 10 15 20 25 30
tan# : 0.00 0.0875 0.1763 0.2679 0.3640 0.4663 0.5774 Find the value of tan 3, tan 16, tan 28 stating the appropriate formula used,
b) Using Runge - Kutta method, find the approximate value ofy (0.2) if
= x + y2 given that y 1 when jc 0 take h 0.1
dx
OR
a) Given
x : 1 1.2 1.4 1.6 1.8 2.0
ffx) : 0 0.1280 0.5440 1.2960 2.4320 4.0
Find : /(LI), /'(1.2), /'(1-8)
b) Use Milnes Predictor - Corrector method to solve the equation
dy dx
= x-y2 at jc = 0.8, given that
dx
y (0) = 0, y (0.2) = 0.02, y (0.4) = 0.0795, y (0.6) = 0.1762.
3E2071 (4)
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Attachment: |
| Earning: Approval pending. |
